We construct a locally compact Hausdorff topology on the path space of a finitely aligned k-graph Λ. We identify the boundary-path space ∂Λ as the spectrum of a commutative C*-subalgebra of C*(Λ). Then, using a construction similar to that of Farthing, we construct a finitely aligned k-graph Λ̃ with no sources in which Λ is embedded, and show that ∂Λ is homeomorphic to a subset of ∂Λ̃. We show that when Λ is row-finite, we can identify C*(Λ) with a full corner of C*(Λ̃), and deduce that is isomorphic to a corner of . Lastly, we show that this isomorphism implements the homeomorphism between the boundary-path spaces.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm204-2-4,
author = {Samuel B. G. Webster},
title = {The path space of a higher-rank graph},
journal = {Studia Mathematica},
volume = {204},
year = {2011},
pages = {155-185},
zbl = {1235.46049},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm204-2-4}
}
Samuel B. G. Webster. The path space of a higher-rank graph. Studia Mathematica, Tome 204 (2011) pp. 155-185. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm204-2-4/